Facet evolution on supported nanostructures: Effect of finite height

Abstract

The surface of a nanostructure relaxing on a substrate consists of a finite number of interacting steps and often involves the expansion of facets. Prior theoretical studies of facet evolution have focused on models with an infinite number of steps, which neglect edge effects caused by the presence of the substrate. By considering diffusion of adsorbed atoms ͑adatoms͒ on terraces and attachment-detachment of atoms at steps, we show that these edge or finite height effects play an important role in the structure's macroscopic evolution. We assume diffusion-limited kinetics for adatoms and a homoepitaxial substrate. Specifically, using data from step simulations and a continuum theory, we demonstrate a switch in the time behavior of geometric quantities associated with facets: the facet edge position in a straight-step system and the facet radius of an axisymmetric structure. Our analysis and numerical simulations focus on two corresponding model systems where steps repel each other through entropic and elastic dipolar interactions. The first model is a vicinal surface consisting of a finite number of straight steps; for an initially uniform step train, the slope of the surface evolves symmetrically about the centerline, i.e., the middle step when the number of steps is odd. The second model is an axisymmetric structure consisting of a finite number of circular steps; in this case, we include curvature effects which cause steps to collapse under the effect of line tension. In the first case, we show that the position of the facet edge, measured from the centerline, switches from O͑t 1/4 ͒ behavior to O͑t 1/5 ͒ ͑where t is time͒. In the second case, the facet radius switches from O͑t 1/4 ͒ to O͑t͒. For the axisymmetric case, we also predict analytically through a continuum shock wave theory how the individual collapse times are modified by the effects of finite height under the assumption that step interactions are weak compared to the step line tension.

Discrete and Continuum Relaxation Dynamics of Faceted Crystal Surface in Evaporation Models

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php related to the facet radius and speed, respectively. The idea of viewing facets as shocks is not new; in [13,14,15,16,52] PDE solutions are connected to shocks in the continuum scale. Here, we aim to connect shock-like solutions to step dynamics.…”

“…In this vein, we exactly evaluate the jump of ξ for M2. This is not the first time facets are connected to shocks; see [12,13,14,15,16,52] where PDE solutions are recognized as shocks. We adopt the shock notion with a different perspective: to link this picture to step schemes.…”

“…Using Lagrangian variables [12,13], we view the step radius, R, as a function of height, h, and t. Let χ = h f (0) − h, which measures the height relative to the initial surface structure in the (physical) domain with h < h f (t), and set R = R(χ, t) in this domain. Downloaded 02/26/13 to 129.2.57.67.…”

“…To solve the (free-) boundary value problem of (6.1)-(6.4), we first apply a transformation of (η, m) that (i) maps (η f , ∞) to a finite interval and (ii) renders linear the highest-order derivative in ODE (6.1) [11,13]. Simply apply a simple translation,…”

Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

“…A natural first step might be to consider the flattening of an infinite groove -another example from [9]. Similar issues arise in the radial setting [5,10]. The paper [11] also raises an interesting modeling question: can one determine, by an atomic-scale or Burton-Cabrera-Frank-type argument, how steps of opposite sign should interact?…”